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Omni-MATH / Let P be a polynomial with integer coefficients such that P(0)=0 and

Problem

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problem

Let PP be a polynomial with integer coefficients such that P(0)=0P(0)=0 and gcd(P(0),P(1),P(2),…)=1.\gcd(P(0), P(1), P(2), \ldots ) = 1. Show there are infinitely many nn such that gcd(P(n)−P(0),P(n+1)−P(1),P(n+2)−P(2),…)=n.\gcd(P(n)- P(0), P(n+1)-P(1), P(n+2)-P(2), \ldots) = n.
Plain-text mathematical notation (without MathML)
Let P be a polynomial with integer coefficients such that P(0)=0 and
gcd(P(0),P(1),P(2),…)=1.
Show there are infinitely many n such that
gcd(P(n)−P(0),P(n+1)−P(1),P(n+2)−P(2),…)=n.
Original LaTeX notation
Let $P$ be a polynomial with integer coefficients such that $P(0)=0$ and
\[\gcd(P(0), P(1), P(2), \ldots ) = 1.\]
Show there are infinitely many $n$ such that
\[\gcd(P(n)- P(0), P(n+1)-P(1), P(n+2)-P(2), \ldots) = n.\]

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Source and history

task_id
3e2c8498-8e66-5ea3-ac80-150b0ac26849
task_key
test--3e2c8498-8e66-5ea3-ac80-150b0ac26849
task_revision_id
5
config / split
/ test

Official source

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